Unitary Harmonic Numbers

As I'm creating this post it is the first day of 2024 but on the last day of 2023, I came across the term Unitary Harmonic Number for the first time. This is not surprising as they are quite rare. The initial numbers, up to 40000, are 1, 6, 45, 60, 90, 420, 630, 1512, 3780, 5460, 7560, 8190, 9100, 15925, 16632, 27300 and 31500. Yesterday, my diurnal age was 27300 which is why the term came to my attention.

A unitary harmonic number is defined as a number whose unitary divisors have a harmonic mean that is an integer. This is clearly not often the case. Let's take the number 12. It has divisors of 1, 2, 3, 4, 6 and 12. Of these, only 1, 3, 4 and 12 are unitary divisors. Let's recall that a unitary divisor of a number is a divisor such that, when divided into the number, the result is a number that has no factors in common with the divisor. For example, 2 divides into 12 to give 6 but 6 and 2 have 2 as a common factor and so 2 is not a unitary divisor. 3 however divides into 12 to give 4. 3 and 4 have no common factor and so 3 is a unitary divisor. 

Let's look at 27300. It has the following divisors:

1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 15, 20, 21, 25, 26, 28, 30, 35, 39, 42, 50, 52, 60, 65, 70, 75, 78, 84, 91, 100, 105, 130, 140, 150, 156, 175, 182, 195, 210, 260, 273, 300, 325, 350, 364, 390, 420, 455, 525, 546, 650, 700, 780, 910, 975, 1050, 1092, 1300, 1365, 1820, 1950, 2100, 2275, 2730, 3900, 4550, 5460, 6825, 9100, 13650, 27300

There are 32 unitary divisors of 27300 and they are:

1, 3, 4, 7, 12, 13, 21, 25, 28, 39, 52, 75, 84, 91, 100, 156, 175, 273, 300, 325, 364, 525, 700, 975, 1092, 1300, 2100, 2275, 3900, 6825, 9100, 27300

The harmonic mean of a set of numbers is defined as the reciprocal of the average of the reciprocals of the numbers. The sum of the 32 reciprocals of the unitary divisors is 32/15 and thus their average is 1/15 which becomes 15 when we consider the reciprocal. Numbers like 27300 comprise OEIS A006086 (permalink):

A006086

Unitary harmonic numbers (those for which the unitary harmonic mean is an integer).

The next unitary harmonic number will occur when I'm 31500 days old which I may or may not be around to celebrate. For posts relating to the harmonic mean see Reciprocals of Primes and Root-Mean-Square And Other Means.